#### A Set of Point Charges

Four charges of equal magnitude are placed at the corners of a square that measures L on each side. There are two positive charges Q diagonally across from one another, and two negative charges -Q at the other two corners.

How much potential energy is associated with this configuration of charges?

1. Zero
2. Some positive value
3. Some negative value

One way to determine the total potential energy is to add up the energy from each interacting pair of charges. There are six pairs to consider. Four pairs have a potential energy given by, for example:

U12 = -KQ2/L

The energy for the two diagonal pairs is of the form:

U13 = +KQ2/(21/2 L)

A second way to answer the question is to re-phrase it as "How much work was required to assemble this set of charges in this configuration?"

Bring the charges in from infinity one at a time.

• It takes no work at all to bring in charge 1.

• Bringing in charge 2 takes negative work, because we have to hold it back since it's attracted to charge 1.

• Bringing in charge 3 takes very little work, since there's already one + charge there and one - charge, and the work done is also negative because it ends up closer to the negative charge.

• Bringing in the fourth charge also takes negative work because there are two positive charges and one negative charge, so overall it's attracted to them.